Graphing Polar Curves: Circles, Cardioids, Roses, and Limacons

A polar curve r = f(theta) is built point by point: pick an angle theta, compute r, and plot that distance from the origin along that direction. As theta sweeps from 0 to 2 π, the point traces a path. Circles like r = a cos(theta) or r = a sin(theta), cardioids like r = 1 + cos(theta), roses like r = cos(n theta), and limacons like r = 1 + 2 sin(theta) all come from this same sweeping process; the differences in shape come entirely from how fast and how often r hits zero or flips sign as theta increases.

For roses, r = cos(n theta) is zero whenever n theta is an odd multiple of π/2, and each time r passes through zero the curve pinches back to the origin and starts a new petal. If n is odd, the curve retraces itself halfway through, so you only get n petals instead of 2n — that's why r = cos(3 theta) makes three petals, not six. Cardioids trace one continuous loop with a single pinch point at the origin, since r touches zero exactly once per revolution. Limacons either have an inner loop (when the constant term is smaller than the coefficient on sine or cosine), a dimple, or no dip at all, depending on that ratio.

This topic sets up polar area and arc length, where you integrate using the same theta bounds you find by locating where r = 0. Knowing where one petal starts and ends, or where a loop closes, is exactly what lets you set correct limits of integration later.